


Spires and Spirit Levels

by Iruka



Category: Frozen (2013)
Genre: Frozen II
Language: English
Status: Completed
Published: 2019-05-13
Updated: 2019-05-13
Packaged: 2020-03-02 16:50:49
Rating: General Audiences
Warnings: No Archive Warnings Apply
Chapters: 1
Words: 2,980
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/18815020
Author URL: https://archiveofourown.org/users/Iruka/pseuds/Iruka
Summary: As the sisters prepare to test Elsa’s powers scientifically, Elsa tries to cure Anna’s math phobia with a challenge.





	Spires and Spirit Levels

“…and this cute little guy is the clinometer.” Queen Elsa grinned as she held up a brass tube that looked sort of like a spyglass, except for the metal half-circle marked with measuring lines attached to the bottom. 

The royal sisters of Arendelle sat in Elsa’s study, beside a desk littered with the piles of papers Elsa had to handle every day. Anna had persuaded her sister to take a break from her usual work and look over a new delivery of scientific measurement tools Kai had brought from town. 

Elsa appeared to know something about the instruments, but Anna found it all new and confusing. Still, confusion was way better than the fear she’d felt a few weeks ago, when she’d caught Elsa charging into a stormy ocean and convinced her to try a less solitary, and hopefully less deadly, way of testing her powers. With Anna along for the ride, of course, as a brake on Elsa’s occasionally worrying self-destructive streak. 

_She’s toned things down because she knows I’d totally get myself killed trying to save her if she got into trouble._ While Anna wished there were an easier way, she’d put herself on the line as often as necessary to keep Elsa safe. 

Anna had gotten herself a little notebook with a water-resistant leather cover for keeping notes on their experiments. Mostly they just made things up as they went, although Elsa, the ultimate math nerd, kept pushing toward more complicated instruments and calculations. The past few weeks had been an amazingly fun, if aimless, spree of attempts to stick numbers on anything either of them could think of to try with Elsa’s powers, from covering the castle’s roof with snow and clearing it again (elapsed time: one minute, 32 seconds) to melting a chunk of the snowcap on North Mountain (roughly half an hour, which would have been longer if Anna hadn’t dropped their stopwatch off the balcony of the Ice Palace, forcing Elsa to stop and dig it out of the snow on the mountainside). 

Elsa waved the clinometer’s little brass arm its hinge as if it were saying hello. She seemed ridiculously pleased with it, as she had been with all the sciencey stuff she’d managed to dig out of castle storage and find in the local stores. The stopwatch was the newest, most up-to-date model Arendelle’s clockmaker had to offer. 

“What’s it for?” Anna asked, absently waving back. 

“You can use it to measure an incline—the steepness of a slope. Say we’re trying to estimate the volume of snow on a mountainside—"

Anna picked up another tool, a long, narrow wooden frame that held a glass tube full of liquid. “Why do they call this thing a _spirit level_? Makes it sound like it’s for measuring ghosts or something.” 

“Not that kind of spirit. It means alcohol.” Elsa tapped the tube with its little suspended air bubble. “In here.” 

“Oh.” Anna bit her lip before asking her next question. “Just to get things straight, Elsa, you’re going to be taking care of all the actual math stuff, right? The _math_ math stuff, the stuff that’s more than handling the stopwatch and writing things down? Because that’s not exactly my strong point.” 

Elsa looked at her. “If you really want to help with measuring the extent of my powers, you’ll need to learn to do geometric calculations too. For one thing, we’ve barely started figuring out the volume of snow I can shift, and how fast. The further into this project we get, the more I’m going to have my hands full with other things. Literally.” 

“Well, couldn’t I just jot down the numbers and let you do the hard part later? I mean, it’s fun for you, right?”

“That isn’t the point.” Her sister raised one eyebrow in a judgy sort of way that made Anna’s palms sweat. “I think a little calculation exercise may be in order.” 

Anna sighed inwardly. _Had to open my big mouth._

Elsa pulled over a bit of paper and jotted some math symbols on it with a quill pen. “Here’s the trigonometric formula for calculating the height of a tall object. I know you wrote the procedure in your notebook a few days ago. Probably thinking I’d never make you use it.” 

Anna shrugged and smiled sheepishly. 

“Come back,” Elsa continued, “when you can tell me the height of the spire on top of the castle to the nearest foot.” 

“Huh? That’ll take me all day to figure out,” Anna protested. “Can’t you think of anything else for me to help with? Some nice, normal paperwork?” 

“This is more important. You’ll be a better observer if you understand the mathematical underpinnings of what we’ll be doing. Besides, I’m not the only one who needs to push my limits. Think of it as a challenge,” she added, seeing the look on Anna’s face. “You enjoy a good challenge, right?” 

_Sure—when it’s not_ boring. Anna held back the words but didn’t try to hide her displeasure as she took the paper from Elsa’s hands and reluctantly headed for the door. 

“Don’t forget this.” Elsa stood up and handed her the clinometer. “And the spirit level.” She piled the second instrument on top of the first. Despite Elsa’s explanations, Anna had only the vaguest idea what she was supposed to do with them.

“Have fun!” With a cheerful smile, Elsa ushered her out the door. Anna was too irritated to smile back. If Elsa meant what she’d said about letting her back in once she’d solved the problem, this door might never open for her again. 

With the instruments and paper in hand, Anna wandered gloomily out to the courtyard and looked up at the castle’s tall central spire, adorned with Elsa’s snowflake. 

She began by spending a few minutes trying to figure an easier way than Elsa’s formula to find the height of the castle’s highest point. She could estimate the height of a row of shingles, count the number of rows up to the spire, and multiply, since she could just about handle multiplication… Or find another building of similar height in the town, and ask the owner how tall it was… Or get a really, really long string, climb up on the roof, and… 

No, none of those methods would be close enough. The right way was supposed to be just as good for a mountain peak as for a spire, and it had to use both of the tools Elsa had given her. 

Sighing, she looked at the paper in her hand. Elsa’s handwriting was clear and elegant, but the meaning behind it was completely opaque to Anna. 

After a few more minutes of thought, she hit on the obvious solution to her problem—cheating. 

First, she went looking for Kai, buttonholing him in the front hall. “Kai, you know a lot about the castle, right? You wouldn’t happen to know how tall—” 

Kai’s face flushed red, and she could actually see his bald head start to sweat. “My apologies, Princess, but Queen Elsa has already given the staff explicit instructions not to assist you with your, um, math test.” 

Anna grumbled under her breath at Elsa for being one step ahead of her, but there was still something left for her to try. She made her way to the castle’s archive room. 

Her father had always been more personally concerned with Elsa’s education than with Anna’s, but sometimes he had also managed to spend one-on-one time teaching his second daughter. During one of those times he had introduced her to the archive, with its rows of shelves holding neatly organized books, folders, and ledgers with documents and information from all over Arendelle. 

Just as Elsa sometimes dropped random math problems on Anna’s head, the king would send her to find, say, the ledger showing total revenues from the kingdom’s ice exports in 1826, or whatever. While she couldn’t say she knew what was in every document in the royal archives, she was familiar with a whole lot of them. 

One of the tall leather folders, she knew, contained a copy of the plans for Arendelle Castle. If only she could remember where it was. 

The staff had kept the archive up to date over the years. There were areas for the records of births and deaths sent in by the kingdom’s towns and villages, for government stuff like taxes, and for letters from foreign dignitaries. She skimmed through several sections before finding what she wanted. 

She pulled the correct folder off the shelf and onto a table, flipped it open, and stopped. There were the plans, hand drawn on parchment with an artist’s skill, every dimension of the castle neatly marked and numbered. 

There was something newer there, as well—a note in Elsa’s handwriting that read: _Do you really want to do this?_

_She’s not one or two steps ahead,_ thought Anna, feeling a little staggered. _More like a dozen._ The note could mean “Do you really want to work together?” or “Do you really want to cheat?” Anna was pretty sure that if Elsa were here to ask, she’d raise an eyebrow and say, “Pick one.” 

She was also sure that if she did cheat off the plans and take the correct number to Elsa right now, her sister wouldn’t call her out. She would probably say nothing more about it. But Anna didn’t think even Elsa would be able to fully hide her disappointment. 

She didn’t want to disappoint Elsa. Not ever. 

Anna huffed in frustration and closed the folder, putting it back on the shelf. Then she went to her own room to get the notebook where she had written the steps she would need to figure out Elsa’s formula. 

Her notes said that to calculate the height of a tall object, she needed to look at its top through the clinometer—after using the spirit level to make sure she was resting it on a flat surface—then measure the distance from that point to the object’s base.

_Great. The spire doesn’t have a base, because it doesn’t touch the ground._

_You think she hasn’t thought of that? What would she say to do?_

Elsa would probably tell her to figure out where it _would_ touch the ground, if it _did_ touch the ground. That was in the castle’s large central chamber—Elsa’s throne room—where you could stand in the middle of the floor and look up into the hollow tower that narrowed at the top into the spire. 

She went there and looked over the room before deciding she would need to find the exact center, then measure to a point in the courtyard where she could use the clinometer. She could try to guess about where the center was by standing beneath the spire and looking up, or by counting floor tiles. 

Or she could do it the right way—the way she knew Elsa intended—by measuring the walls themselves and marking their centers on the floor, then extending those lines until they crossed beneath the spire. She’d need some kind of rope or string to make a measuring line, then she’d have to mark off every foot… 

It sounded like an awful lot of work. 

She sighed. _Better get started if I want to finish today._

She turned toward the door to find Kai standing there, holding something out to her. “The queen instructed me to give you this once you were ‘clearly on the right track.’ I believe that’s now, ma’am.” 

She took the measuring tape from his gloved hands, smiling her thanks. It would cut her workload in half. 

Even with its help, it took her more than an hour to finish mapping out the center of the ballroom floor, then measuring from there to the courtyard, where she could see the top of the spire through the brass tube and mark its angle of elevation with the little swinging arm of the clinometer. 

Then came the actual calculations. Wrapping her mind around those was a slow, frustrating process, like chipping away ice with a toothpick—except the ice was inside her skull, and chipping it was _exhausting._

Another hour crawled painfully by before she felt certain she had the right result and satisfied that she could explain to Elsa how she’d gotten it. When she entered Elsa’s study again, her sister did not appear to have moved from her place behind the desk; only the heights and positions of the paper piles around her had changed. 

“Elsa?” 

Elsa blinked, looked up at Anna, and leaned back in her chair, smiling tiredly. “How’d it go?” 

Anna shrugged. “It went.” She laid her single sheet of paper on the desk. 

Elsa picked it up and examined it. “Good answer. Nicely done.” 

Anna felt a warm rush of pride at her sister’s praise. 

Then Elsa laid down the paper and met Anna’s eyes, raising an eyebrow as though waiting for something more. 

“Okay,” Anna sighed, “yes, I tried the archive and found your note. Happy?” 

“Oh, I’m not judging you.” Elsa laced her fingers together, smiling her gentle smile. “In fact, you may be interested to know I tried the same trick when Father gave me the spire problem.” 

“You _what?_ ” 

“It’s true. I got the worst lecture ever about cheating when he found out. Then he made me do it right, and I learned that I actually enjoyed math. It was quite a turning point for me. An epiphany.” She looked at Anna hopefully. “I don’t suppose you…” 

“Um.” Anna tucked a strand of hair behind her ear and gave Elsa an apologetic little smile. “Nope. Sorry.” 

“Well.” Elsa shrugged. “Anyway, you solved it faster than I did. It took me three days.” 

“Wow.” Anna felt a grin spread over her face. “I beat your time? On a _math problem_?” 

“Yeah, you did. Of course, I was nine when I did it.” 

“Oh.” 

Anna took a closer look at the papers covering the desk, and realized that they were not, as she’d assumed, more diplomatic correspondence or notes on international treaties. 

She picked up one of the papers, a half-finished bulleted list in Elsa’s handwriting titled “Fluid Dynamics,” and her eyes strayed over others—“Melting Rates,” “Snow-to-Water Ratios,” “Ice Classifications and Characteristics,” sketches of the local glacier marked with numbers, a map labelled “Arendellian Watersheds”… 

“This isn’t your usual queen stuff,” she observed. 

Elsa gave her a guarded look. “I finished that hours ago. This is just… playing around.” 

“Really?” Anna tugged on the paper beneath Elsa’s tapping index finger, which had “Notes on Cryokinetic Terminology” written across the top, and Elsa reluctantly let her take it and skim over it. Anna thought she grasped, dimly, what her sister was trying to do—combine what they learned about her powers with the broader ideas laid out on the desk in front of her. “Looks to me like you’re inventing a whole new branch of science in your spare time.” 

That was going to be an awfully big job. She’d need a lot of help. 

Elsa sighed. “I knew you’d try to make too big a deal out of it.” 

“Elsa, you realize you’re a genius, right?” When this remark got her a pained look, Anna added, “I’m going to make you admit it someday. Call it a life goal.” 

“I’m just gathering information, that’s all. I have no idea whether anything useful will come of it.” 

“Me neither,” said Anna, “but maybe we can figure out a way to use it. I might’ve seen something helpful in the archive room…” 

An idea was sprouting in her mind. In Arendelle—stony and green in summer, buried under piles of ice and snow for most of the other three seasons—winter claimed lives every year. Maybe it didn’t have to be that way. 

The day’s work had led to an epiphany of sorts, Anna thought as she headed back to the archives to dig up some numbers of her own. She had decided that if her sister needed her to be good at calculating the slope of a hillside or the volume of a snowdrift, she no longer cared whether or not she enjoyed it. If she hadn’t learned to love numbers, she _had_ learned that she couldn’t let hating their guts stop her from mastering them to help Elsa. 

That was just what a good right hand did. 

“Okay, math,” she said under her breath as she smacked one fist into the opposite hand, “prepare to have your ugly face pounded in by Princess Anna of Arendelle!” 

She slammed open the door of the archive room and charged into the death records section, not certain what she was looking for except that she would know it when she saw it. 

*****

An hour later, Elsa came to the door to ascertain why Anna’s archival research seemed to require so much banging and yelling. 

“Anna?” she called with a perfunctory knock, and opened the door to a scene of chaos—at least one entire section of documents pulled from the shelves and spread over every table and a good part of the floor, chairs piled in an alarmingly unstable fashion along the wall—before Anna leaped into the doorway, blocking her view and forcing her to retreat a step. 

“You know how to replace all that?” she asked skeptically. 

Anna waved a dismissive hand. “Of course I know. Archive and I go waaay back.” She seemed a little breathless. “Definitely on the right track here—finding some amazing stuff. Gonna make you proud, Elsa!” 

Before Elsa could answer “You already have,” her sister gave her a grin and a thumbs-up, slammed the door in her face, and was gone. 

Elsa rubbed the headache spot between her eyes and murmured, “So that’s what that feels like.”

**Author's Note:**

> This oneshot started out as a flashback in the middle of a much longer story, but then I realized how easily it could be extracted whole. (It was getting a bit long for a flashback anyway.) It’s basically a sequel to my one previous Frozen fic, “The Magic One,” which means it occurs sometime after the first scene of the Frozen II teaser trailer. Reading that story is not necessary in order to understand this one, but that doesn’t mean I wouldn’t appreciate it if more people did!


End file.
